I don't think there is an accepted definition of a "black dwarf" - it is not a term used in the scientific literature.
A popular definition that appears to circulate on the internet is that it is a white dwarf that has cooled down to the extent that it no longer emits any radiation in the visible part of the spectrum. But this is an unworkable theoretical definition. Even the coolest objects emit some radiation in the optical part of the spectrum; whether or not it is detectable will depend on the size of the object and how far away it is from us.
If a black dwarf is simply to be given the practical definition of being unobserved/unobservable at visible wavelengths then such objects exist already. Kaplan et al. (2014) have found evidence for a $\sim 1M_{\odot}$ white dwarf companion to a pulsar that they are unable to detect at visible wavelengths even with the Keck telescope down to an absolute (red) magnitude of $M_R>19.1$.
They argue that this means the white dwarf has cooled below 3000 K. This is possible within the age of the universe because more massive white dwarfs cool more rapidly because their higher interior densities lead to more rapid crystallisation and a steep decrease in their heat capacity when they enter the Debye cooling regime. A more typical old white dwarf of mass $\sim 0.6M_{\odot}$ would still be at a temperature of $\sim 4000$K and emit a reasonable amount of light at visible wavelengths.
Thus, if you are happy to accept the definition above, then black dwarfs can result from the cooling of massive white dwarfs in about 10 billion years. The plot below, taken from the paper of Kaplan et al. (2014), shows some cooling models for massive white dwarfs with hydrogen atmospheres. They easily cool below 3000 K within 10 billion years.
To address Mark Foskey's point: Once in the Debye cooling regime, the cooling rate hardly slows because the heat capacity is $\propto T^3$, whilst the luminosity is roughly $\propto T^{3.5}$ (here, $T$ is the isothermal interior temperature). Luminosity is also $\propto T_s^4$, where $T_s$ is the surface temperature.
The relevant differential equation is that the rate of change of thermal energy equals (minus) the luminosity
$$ \frac{d}{dt}\ \int T^3\ dT \propto -T^{7/2}$$
Putting this together, we find that $dT_s/dt \propto -T_s^{3/7}$ and so the cooling rate does not change much once Debye cooling starts. As a consequence the temperature plummets as $T_s \sim -t^{7/4}$ and the time taken to cool well below 1000K from 3000K would only be a few times longer (maybe a billion years) than the time to cool from 4000K to 3000K (as suggested by a simple by-eye extrapolation of the models shown in the plot).
The same treatment in the "Mestel" cooling regime, where the heat capacity is constant yields a much shallower $T_s \sim t^{-7/20}$, and accounts for the sharp "knee" at the onset of Debye cooling.
